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Version Date: June 26, 2010 Derivation of the Feynman From Chapter 3 of Student Guide to Quantum Theory, by Robert D. Klauber ©

3.0 The Scalar Feynman Propagator The Feynman propagator, the mathematical formulation representing a , such as the one represented by the wavy line in Fig. 1-1 of Chap. 1, is the toughest thing, in my opinion, to Feynman propagator not learn and feel comfortable with in QFT. If you don’t feel comfortable with it right away, don’t simple to understand worry about it. That is how virtually everyone feels. Over time, it will become more familiar, and if you are lucky and work hard, maybe even easy. Use wholeness chart as I have tried to take the derivation of the propagator one step at a time, and emphasize what each you study the derivation step entails. Wholeness Chart 5-0X (also at Free Fields Wholeness Chart link at www.quantumfieldtheory.info) breaks these steps out clearly, and should be used as an aid when studying the propagator derivation. : NRQM vs QFT and Real vs Virtual Particles Note that the propagator for real particles, which you may have studied in NRQM, is not the Feynman propagator for same as the Feynman propagator, which is explicitly for virtual particles in QFT. It may be QFT virtual particles is confusing, but the Feynman propagator is often simply called, “the propagator”. You will have to different from propagator get used to discerning the difference from context. for real particles of NRQM & RQM In QFT, as we will see when we study interactions, a propagator for real particles is not generally needed, and we will not derive one here. 3.0.1 The Approach The first part of QFT is a free particle theory (no interactions, as in this chapter and the next We’ll use the Feynman three). After this, interactions are introduced. In the course of deriving the interaction theory, a propagator when we get mathematical relationship arises that is called the Feynman propagator. Physically, it can be to interaction theory visualized as representing a virtual particle that exists fleetingly and carries energy, momentum, and in some cases, charge from one real particle to another. Thus, it is the carrier, or mediator, of force (interaction.) See the virtual of Fig. 1-1 in Chap. 1. It will help us pedagogically to derive the Feynman propagator now, rather than when we get to But it’s easier in the long interactions. The derivation of interaction theory is fairly complicated and it will be easier, as we run if we derive it here develop it, if we already know the mathematical relation for the Feynman propagator, rather than diverting our attention for several pages to derive it then. Heuristically, it may help to consider the virtual particle as created at a particular point and destroyed at a later spacetime point, and this is how Feynman diagrams portray it. From † this (heuristic) perspective the operator field φ (y) can be considered to create a virtual scalar particle at event y (we used the symbol x2 in Fig. 1-1), and the field operator φ (x) destroys that virtual particle at event x (x1 in Fig. 1-1.) The scalar propagator incorporates these two field operators in a sort of “short-hand” way. Note that the above “creation/destruction at a point” perspective can help initially in understanding the derivation of the propagator, but we caution that it will have to be modified and

39

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refined. We will save that to the end when, after digesting the derivation to follow, this modification will be easier to understand. We will now derive a relationship for the propagator using the field operators acting on the , and will later see (Chap. 7? XXX) that this derived relationship arises naturally in the full mathematical development of the interaction theory. 3.0.2 Milestones in the Derivation We start with the coefficient commutation relations and proceed in five distinct steps. The entire Start with the coefficient derivation is for continuous (not discrete) eigenstate solutions of the field equation (Klein-Gordon and follow here), since the propagator represents a virtual particle in the vacuum and the vacuum is not 5 distinct steps confined to a volume V. We represent the scalar Feynman propagator with the symbol ∆F (x − y) . As noted, our starting point is the coefficient commutators for continuous solutions, Use continuous solutions † † » a (k) ,a (k′)ÿ⁄= » b (k) ,b (k′)ÿ⁄= δ (k − k′) (continuous) , (3-1) to field equation and from there we follow five steps. The first two are purely mathematical, and will serve as background that will help us with the remaining steps. The third step comprises a physical interpretation of the propagator, and the last two steps are mathematical manipulations to get that interpretation into convenient form (the form that is used in QFT interaction analysis.) Step 1: Use (3-1) to find commutation relations for positive and negative frequency solutions Overview of the 5 steps (see (3-37) in Chap. 3), i.e., determine » ± † _ ÿ ± φ (x) ,φ ( y)⁄= i∆ (x − y) , (3-2) where i∆± (x − y) is the symbol representing the two commutators on the LHS of (3-2). i∆+ represents the upper + and – signs on the LHS; i∆– represents the lower – and + signs. ± ∆ is slightly different from, and will be used to find, the Feynman propagator ∆F. Step 2: Express i∆± (x − y) in terms of a contour integral in the complex plane.

Step 3: Express the Feynman propagator ∆F as a mathematical representation of a particle or created at one point in space and time in the vacuum and destroyed at another place and time. ± Step 4: Use i∆ (x − y) of Step 2 to express ∆F as a contour integral in the complex plane.

Step 5: Re-express ∆F as an integral in real (rather than complex) space. This is the form most suitable for analysis. 3.0.3 The Derivation Step 1: Commutation Relations for Positive/Negative Frequency Solutions Define the symbol i∆+ as the of the field type a solutions, i.e., Step 1, first part, + » + †− ÿ + i∆ (x − y) = φ (x) ,φ ( y)⁄, (3-3) find i∆ = commutation relation for type a fields where the solutions used on the RHS are the integral (continuous) form for the Klein-Gordon solutions (see (3-37) in Chap. 3). It is common usage to use a + sign to designate (3-3), rather than the letter a, which would be easier to remember. Just think “a type field” when you see +. Equation (3-3) is thus ikx ik y 1 e− e ′ + »a ,a† ′ ÿ 3 3 ′ i∆ (x − y) = 3 —— (k) (k )⁄ d k d k 2 (2π ) ωkωk′ (3-4) ik y 1 ≈ e ′ ’ ∆ δ − ′ d 3 ′÷e−ikxd 3 , = 3 — ∆— (k k ) k ÷ k 2 (2π ) « ωkωk′ ◊ and hence, Section 3.0 The Scalar Feynman Propagator 41

−ik ( x− y) + 1 e 3 i∆ (x − y) = 3 — d k . (3-5) 2( 2π ) ωk Similarly, where a minus sign stands for b type fields (since they are associated with , the minus makes some sense), Step 1, second part, ik y 1 eikxe− ′ – − » − †+ ÿ » † ′ ÿ 3 3 ′ find i∆ = commutation i∆ (x − y) = φ (x) ,φ ( y)⁄= 3 —— b (k) ,b (k )⁄ d k d k 2( 2π ) ω ω relation for type b fields k k′ (3-6) ik ( x− y) −1 e 3 = 3 — d k , 2( 2π ) ωk which leads to

_ik ( x− y) ± 1 e Step 1, final part, i ± x y » ± x , †_ y ÿ d 3 (3-7) ∆ ( − ) = φ ( ) φ ( )⁄= 3 — k combine above parts into 2 (2π ) ωk one symbol i∆±

± Ste p 2: Expressing the Two i∆ as Contour Integrals

Consider the complex plane for a function f of the Review of integral in complex variable k0, i.e., f (k0). Here, the symbol k0 is not a pole (poles are usually designated with null the complex plane subscript), but represents a complex number generalization of the zeroth component (the energy) of 4-momentum k. We concern ourselves with the particular case where k0 takes on the real value ωk. Figure 3-3. Contour Integral From complex variable theory, for Real, Positive Frequency

1 f (k0 ) f (ωk ) = dk0 . (3-8) i2π — k C+ 0 −ωk Now, re-express (3-5) as

À −iωk (tx −ty ) ¤ Step 2, first part, 1 ikY x−y Œe Œ + ( ) d 3 + i∆ (x − y) = 3 — e à ‹ k , (3-9) express i∆ as a (2π ) Œ 2ωk Œ contour integral %Õ ((&(('›

f (ωk ) where we take the bracketed quantity as equal to f (ωk), and where −ik t −t e 0 ( x y ) f (k0 ) = . (3-10) k0 + ωk

We can then use (3-10) in (3-8) to re-express f (ωk) in terms of a contour integral. Using this for the bracket in (3-9), we find (3-9) becomes 4 2 0. Error! No text of specified style in document.

À ¤ + 1 ikY(x−y) Œ 1 f (k0 ) Œ 3 i∆ (x − y) = e à dk0 ‹d k 3 — i2π — k − ω (2π ) ÕŒ C+ 0 k ›Œ

À −ik0 (tx −ty ) ¤ 1 ikY(x−y) Œ 1 e Œ 3 = e à dk0 ‹d k (3-11) 3 — i2π — k − ω k + ω (2π ) ÕŒ C+ ( 0 k )( 0 k ) ›Œ −ik x− y −i e ( ) = d 4k . 4 — 2 2 (2π ) C+ (k0 ) − (ωk ) where the integral notation now implies integration over four dimensions of the 4-momentum, with the 3-momentum part from – ∞ to +∞ in real space and the energy part a contour integral in complex space. Note that the integral does not “blow up” because k0 ≠ ωk over the contour integral. We are using a mathematical trick that works, though it jars our usual understanding that, for real particles, the zeroth component of 4-momentum equals energy. k0 has at this point become, for us, a variable that generally does not equal energy ωk. We modify (3-11) a little by noting what is always true mathematically for any four vector, and thus true for 4-momentum components, Modifying terms in our result a little 2 2 2 2 2 2 k = (k0 ) − (k) → (k0 ) = k + (k) (3-12) and what is physically true for rest mass, energy, and 3-momentum, 2 2 2 2 2 2 ωk − (k) = µ → ωk = µ + (k) . (3-13) Substitute the RH expressions of (3-12) and (3-13) into the last line of (3-11) to get −ik x− y −i e ( ) i∆+ (x − y) = d 4k . (3-14) 4 — k 2 2 (2π ) C+ − µ

Im k 0 For i∆– (x – y), we carry out similar steps except that the contour integral Step 2, second part, – (still c.c.w.) is now about – When express i∆ as a C- ωk. all is said and done, we find the only contour integral difference from (3-14) to be the Re k _ 0 contour, which is now about the ωk negative frequency value and designed by C –. Figure 3-4. Contour Integral for

Real, Negative Frequency

−ik x− y −i e ( ) i∆− (x − y) = d 4k . (3-15) 4 — k 2 2 (2π ) C − − µ Step 3: The Feynman Propagator as the VEV of a Time Order Operator Fig 3-5a represents creation of a particle, which will be virtual, at y and destruction of it at x. Figure 3- 5b represents creation of an antiparticle at x and destruction of it at y. Virtual particles are never Step 3, first part, detected when real particles interact, so the same effect defining the time ordered on the real particles could be realized by either of the operator T and seeing processes in Figure 3. For example, a virtual particle how it represents carrying charge from y to x would represent the same creation of either a charge exchanges as an antiparticle carrying opposite virtual particle or antiparticle followed by its destruction

Figure 3-5. Creation & Destruction of Virtual Particle/Antiparticle Section 3.0 The Scalar Feynman Propagator 43

charge from x to y. Thus, we need a relationship for the propagator that includes both scenarios as possibilities.

That is, we need an operator that will create a particle first if ty < tx, but create an antiparticle first if tx < ty. Our Klein-Gordon solutions of Chap. 3, (3-37), (3-84), and (3-85), provide the means for the desired creation and destruction operations. But these have to be arranged to provide us with the time ordering dependence of Fig. 3-5. To this end, consider the time ordered operator T, defined as follows. † If ty < tx, φ (y) operates first (creates a particle) and is placed on the right, with φ (x) operating second (destroys the particle) and placed on the left. † † for ty < tx T {φ (x)φ ( y)} = φ (x)φ ( y) . (3-16)

† If tx < ty, φ (x) operates first (creates an antiparticle) and is placed on the right with φ (y) operating second (destroying the antiparticle) and placed on the left. † † for tx < ty T {φ (x)φ ( y)} = φ ( y)φ (x) . (3-17) Step 3, second part, We now define what is called the transition amplitude density, which equals the vacuum defining the transition expectation value (VEV) of the above time ordered operator. It is an amplitude, similar to the amplitude density as amplitude of a wave function in NRQM, because, as we will shortly see, the square of its magnitude equal to the VEV of the equals the probability density of it being observed. (As the square of the magnitude of the amplitude time ordered operator T for a component of the wave function equals the probability of it being observed.) This transition amplitude density is † 0 T {φ (x)φ ( y)} 0 , (3-18) We use the VEV because we will be interested in and this represents both possible scenarios of Fig. 3-5. In wave mechanics, the bracket above is an the expectation of finding integration over all space. This is still true, but note carefully that the integration variable is over the a virtual particle space variable of the bra and ket (think x′), but not the time ordered variables x and y. In QFT notation, we tend to merely think of a bracket as equaling zero unless the bra and ket represent the traveling in the vacuum same state. To gain insight into (3-18), consider the transition amplitude density operating on the vacuum when a virtual particle is propagated. Then, where an overbar in a state represents an antiparticle, Gaining insight into the time ordered operator T † † T {φ (x)φ ( y)} 0 = φ (x)φ ( y) 0 acting on the vacuum ≈ ’≈ ’ ∆ ÷∆ ÷ = ∆φ + (x) + φ − (x) ÷∆ φ†+ ( y) 0 + φ†− ( y) 0 ÷ ∆%&' %&' ÷∆ %(&(' %&' ÷ (3-19) ∆destroys creates ÷∆destroys antiparticle, creates ÷ «particle antiparticle ◊« annihilates vacuum particle ◊ = (φ + (x) +φ − (x)) F ( y) φ = G (x) F ( y) 0 + H (x) F ( y) φφ . G, F, and H are numeric factors that result from the creation and destruction operations (such as the normalization coefficients that are part of the field operators), which we will not express explicitly here. Thus, we have a general ket left, which in this case is part vacuum state, with the amplitude of the vacuum state part being GF, and the amplitude of the multiparticle state (scalar plus anti-scalar) part being HF. For appropriate normalization, the square of the magnitude of GF is the probability of observing the vacuum (no particles left after the transition.) To find the amplitude GF, we need only form an inner product of the last line of (3-19) with ¿0Œ, i.e., 4 4 0. Error! No text of specified style in document.

0 T {φ (x)φ† ( y)} 0 = 0 G (x) F ( y) 0 + 0 H (x) F ( y) φφ Taking the inner product of the above T|0≈ with 0 = G (x) F ( y) 0 0 + H ( x) F ( y) 0 φφ = G (x) F ( y) . (3-20) k %&' to get the transition =1 =0 amplitude density Thus, the VEV of the time ordered operator is an amplitude, the square of whose magnitude is 2 the probability of the transition from vacuum initially to vacuum finally. Actually, |G(x)F(y)| is a The square of the probability density (to be precise, a double density), because it is a function of x and y. That is, the absolute value of the location y where the virtual particle is created could be anywhere, and so could the location x where transition amplitude it is destroyed. We would need to integrate the probability density over all possible x and all density is a probability possible y to get the actual probability, and this is what one does in interaction theory to calculate density (for the transition probabilities and cross sections to occur) In a similar way, the same time ordered operator can be used for antiparticle propagation (with time for x and y reversed) as in Fig 3-5b and (3-17). You can prove this to yourself by doing Prob. 17 of Chap. 3. Given all of this, we can define our mathematical relationship for the processes shown in Fig. 3- Redefine the transition 5 as the VEV of the time ordered operator T. This is called, in honor of its discoverer, the Feynman amplitude density as the propagator ∆ (which, by convention, actually includes an extra factor of i), F Feynman propagator † i∆F (x − y) = 0 T {φ (x)φ ( y)} 0 . (3-21) ± Step 4: Expressing i∆F as the Contour Integrals i∆ (x-y) Note what the Feynman propagator equals for ty < tx, the case for a virtual particle (not Step 4, expressing antiparticle). Feynman propagator as two contour integrals † i∆F (x − y) = 0 φ (x)φ ( y) 0 = 0 φ + (x)φ†+ ( y) 0 + 0 φ + (x)φ†− ( y) 0 + 0 φ − ( x)φ†+ ( y) 0 + 0 φ − (x)φ†− ( y) 0 (3-22) %(&(' %(&(' %(&(' %(&(' Step 4, first part, =0 =0 = φ = φ expressing Feynman = 0 φ + (x)φ†− ( y) 0 + 0 φ − (x) φ = 0 φ + (x)φ†− ( y) 0 . propagator for virtual %((&((' particle (not antiparticle) = 0 φφ =0 To the last part of (3-22), we can add zero in the form of 0 = 0 −φ†− ( y)φ + (x) 0 . (3-23) By adding a term equal to %(&(' zero, we can use a =0 commutator we derived Doing that, and using (3-3) of Step 1 to get the second line below, (3-22) becomes in Step 1 + †− †− + » + †− ÿ i∆F (x − y) = 0 φ (x)φ ( y) −φ ( y)φ (x) 0 = 0 φ (x) ,φ ( y)⁄ 0 = 0 i∆+ x − y 0 = i∆+ x − y 0 0 (3-24) %((&(') ( ) numeric = i∆+ ( x − y) .

In similar fashion, for tx < ty, the case for a virtual antiparticle, one finds, by doing Prob. 18 of Step 4, second part, Chap. 3, that expressing Feynman propagator for virtual » †+ − ÿ i∆F (x − y) = 0 φ ( y) ,φ (x)⁄ 0 antiparticle (3-25) = −i∆− (x − y) . In summary, Step 4, summary + ∆F (x − y) = ∆ (x − y) if ty

± where in Step 2, we expressed ∆ (x – y) as contour integrals, and so here we express the Feynman propagator in terms of contour integrals. Note that the Feynman propagator, although encompassing operators in its initial definition, turns out to be simply a numeric quantity without operators.

Step 5: Re-express ∆F in Most Convenient Form - We would like two things more: 1) express ∆ Im k0 the propagator as a single function so we don’t Step 5, expressing have to keep track (while we are integrating Feynman propagator as over spacetime and doing other things) of real, not complex whether the virtual field is a particle or + – (contour), integral − ωk antiparticle (i.e., whether to use the ∆ or ∆ Re k0 function), and 2) have all our integrations over ωk real numbers rather than deal with contour integrals. −∆+ To do this, consider Figure 3-6, where we have shown two contour integrals. The top loop – – + represents ∆ and encloses –ω with a Figure 3-6. Contour Integrals for ∆ and – ∆ k ccw path. The lower loop encloses +ωk, + but since it has a cw integration path, represents – ∆ .

Thus, we can define the Feynman propagator ∆F of (3-26) as proportional to the negative of the integral of Fig. 3-6 for either loop. We say “proportional” because we also have to include the concomitant integration over the 3D space of k not shown in Fig. 3-6, as well as the various constants involved. So we can then re-write the Feynman propagator (3-26) with (3-14), (3-15), and Fig. 3-6, as Two different contours −ik( x− y) for the Feynman 1 e x y d 4k propagator written with ∆F ( − ) = 4 — 2 2 , (3-27) (2π ) k − µ same integral, different CF meaning for path CF where the CF on the integral defines the route we take in the plane of Fig. 3-6. Now, consider enlarging the outer hemispheric parts of the two loops in Fig. 3-6, so they extend essentially to infinity. The value of the contour integrals over them will remain unchanged. But the 2 Extending all parts of k value in the denominator of (3-27) will become so large that any contribution to the integral over those parts of the path will become negligible. This means that we can effectively take the integral contours to Ç except of (3-27) as extending only along the real axis from – Ç to +Ç as in Fig. 3- 7. along real axis

Im k 0 Im k 0

k0 = − (ωk − iη)

– ∞ + ∞ − ωk ω k Re k 0 Re k 0

k0 = ω − iη k Fig. 3-7. Contour CF for ∆F Fig. 3-8. Contour and Displaced Poles for ∆F Instead of integrating We can further simplify by moving the poles an infinitesimal distance η off the real axis as around poles on the shown in Fig. 3-8 and deform the contour so that it is all along the real axis. In the limit as η → 0, axis, move poles slightly off the axis 4 6 0. Error! No text of specified style in document.

the integral will have the same value, though we must now include this slight pole shift in the propagator expression (3-27). We do this by recalling from (3-12) and (3-13) that we used

2 2 2 2 k − µ = (k0 ) − (ωk ) (3-28) to obtain the denominator of (3-27), so we must temporarily restate (3-27) using the right hand side of (3-28), then shift the poles. Thus, (3-27) becomes +∞ −ik(x− y) 1 e 4 ∆F (x − y) = 4 — 2 2 d k . (3-29) (2π ) −∞ (k0 ) − (ωk − iη) If we then use (3-28) again, ignore second order terms in η, and take ε = 2ηωk, we have our final result for the Feynman scalar propagator Yields a single, real integral representing +∞ −ik( x− y) 1 e x y d 4k . both virtual particle ∆F ( − ) = 4 — 2 2 . (3-30) (2π ) −∞ k − µ + iε and antiparticle, the most convenient form Note the advantages of this form. We now have a single mathematical relationship that for the Feynman automatically describes both a particle propagating from y to x and an antiparticle propagating from propagator x to y. We also have done away with the cumbersome contour integrals in favor of a simple 4D integral over the entire 4-momentum space. In practice, we can evaluate this integral then take ε to zero after the integration is carried out. 3.0.4 Comments on the Propagator and Its Derivation The Propagator and Interaction Theory The derivation above was formulated with an eye to interaction theory. In that theory, Our definition of amplitudes are derived for various kinds of interactions between various particles. The square of the Feynman propagator magnitude of each amplitude turns out to be the probability of that particular interaction (transition) here will pop up in our occurring. These transition amplitudes each depend on the initial real particles, the final real formal derivation of particles, and the virtual particle(s) that mediate the transition. It turns out that the factor in the interaction theory amplitude representing the virtual particle contribution is identical to the Feynman propagator ∆F as we defined it in the VEV of the time ordered operator (3-21). Thus, it is also equal to (3-30), so we can simply plug the RHS of (3-30) into the overall transition amplitude as part of our analysis. This is one reason we started with the relation φφ† to create and destroy a virtual scalar particle, rather than what one might initially expect, the simpler creation and destruction operator relation a(k)a†(k). Our heuristic approach was tailored to match what we knew would be coming in the mathematical development of interaction theory.

Meaning of Spacetime Points y and x Feynman diagrams, In Fig. 3-5, we imply the virtual particle is created at y and destroyed at x. In Feynman diagrams and our derivation, virtual particles are depicted in this way, and at least one real incoming particle can be thought of as imply creation and being destroyed at y, as in Fig. 1-1 of Chap. 1, with a virtual particle created simultaneously at y. At destruction at a point, x the virtual particle is destroyed, with the simultaneous creation of at least one outgoing real but more properly, particle at x.. waves are created and To be precise, it is more correct to think of the incoming, outgoing, and virtual particles as destroyed, and they are moving waves spread out in space. What we calculate for a given y and x is the probability density spread out over space. for the interaction as a function of the coordinates y and x. If y and x are closer, one would find the probability density for the interaction to occur is greater; if farther away, the probability density is We are really finding less. Integrating over all x and y gives the total probability for observing the interaction. probability density as Momentum Space Form of the Propagator functions of x and y From (3-30), we can readily write down the 4-momentum space form of the propagator, the of (3-30), which will be very useful, Momentum space form 1 of the propagator ∆F (k) = . (3-31) k2 − µ2 + iε Earlier version was physical space form